My textbook says: Let $R$ denote the set of infinite–by–infinite, row– and column–finite matrices with complex entries. Show that $R \cong R \oplus R$ as $R$–modules.
So for $A, B \in R$, I tried $(A, B) \to A + B, (A, B) \to A, (A, B) \to \left( \begin{array}{cc} A & 0 \\ 0 & B \end{array} \right), $ and $(A, B) \to C$ where the odd-numbered rows of $C$ are from $A$ and the even-numbered rows of $C$ are from $B$ but none of those turned out to be an isomorphism by my calculations and I'm pretty sure the third one isn't even well-defined. How can I exhibit an isomorphism between $R$ and $R \oplus R$?
The described method where you interleave the rows gives a well-defined bijection, because a column has only finitely many nonzero entries exactly when it has both only finitely many nonzero odd entries and only finitely many nonzero even entries.
The issue is that this bijection does not respect the action of $R$, assuming that the action is on the left. This is fixed by doing the same construction with odd and even columns instead of rows.